Introduction

Piessens, Robert; Doncker-Kapenga, Elise de; Überhuber, Christoph; Kahaner, David K.

doi:10.1007/978-3-642-61786-7_1

Cited by 46 publications

(11 citation statements)

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“…The severity pdf, given by ( 9) or (10), has a single mode, which means that the oscillatory nature of the integrand only comes from the sin( ) or cos( ) functions. This well-behaved weighted oscillatory integrand can be effectively dealt with by the modified Clenshaw-Curtis integration method, see Clenshaw and Curtis (1960); Piessens, Doncker-Kapenga, Überhuber and Kahaner (1983). In this method the oscillatory part of the integrand is transferred to a weight function, the non-oscillatory part is replaced by its expansion in terms of a finite number of Chebyshev polynomials, and the modified Chebyshev moments are calculated.…”

Section: The Forward Integrationmentioning

confidence: 99%

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Computing tails of compound distributions using direct numerical integration

Luo¹,

Shevchenko²

2009

JCF

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An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly with little extra effort. High precision results of the 0.999 quantile and CVaR were obtained for compound losses with heavy tails and a very wide range of loss frequencies using the DNI, Fast Fourier Transform (FFT) and Monte Carlo (MC) methods. These results, particularly relevant to operational risk modelling, can serve as benchmarks for comparing different numerical methods. We found that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than MC and competitive with FFT in computing high quantiles and CVaR of compound distributions in the case of moderate to high frequencies and heavy tails.

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Section: The Forward Integrationmentioning

confidence: 99%

“…I ~ or T I can be accurately computed by a IMSL function using the modified Clenshaw-Curtis integration method (Clenshaw and Curtis (1960); Piessens, Doncker-Kapenga, Überhuber and Kahaner (1983)) as described in Section 3.1. ) the one-point approximation is very close to the exact semi-infinite tail integration.…”

Section: Examplementioning

confidence: 99%

Computing tails of compound distributions using direct numerical integration

Luo¹,

Shevchenko²

2009

JCF

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“…4]. Transformed target cells are selectively afTected by the cytotoxic effect of macrophages in most in vitro systems allowing direct macrophage-target cell contact [13,24], and similar selectivity has been described tor some of the soluble cytostatic or cytolytic factors [1,5].…”

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confidence: 99%

Soluble Cytostatic Factor(s) Released from Human Monocytes

Hammestrøm¹

1982

Scand J Immunol

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Human monocytes activated with lymphokines after 3-7 days of in vitro maturation released stable cytostatic factor(s) (CF), as determined by reduction of [methyl-3H]thymidine incorporation and cell counts in target cell cultures. Lipopolysaccharide from Escherichia coli slightly enhanced the release of CF. The CF release was greatest at an intermediate stage of the in vitro monocyte maturation to large, macrophage-like cells. Three transformed human cell lines (NHIK 3025, K-562, and U-20S) and normal skin fibroblasts were consistently inhibited by CF. Normal human mesothelial cells were variably affected, while normal human lymphocytes proliferating in response to allogeneic cells in mixed lymphocyte culture were not inhibited. Further analysis of CF may disclose one of the mechanisms whereby human monocytes and macrophages inhibit growth of human tumour cells.

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“…Statistical inference based on the above likelihood function is potentially computationally intensive. A non-adaptive Gauss-Kronrod integration can be employed to numerically calculate the integral [ 24 ]. The maximum likelihood parameter estimates can be obtained using the Expectation–Maximization (EM) algorithm and Newton-Raphson approximation [ 25 ].…”

Section: Methodsmentioning

confidence: 99%

Revisiting methods for modeling longitudinal and survival data: Framingham Heart Study

Ngwa

Cabral

Cheng

et al. 2021

BMC Med Res Methodol

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Background Statistical methods for modeling longitudinal and time-to-event data has received much attention in medical research and is becoming increasingly useful. In clinical studies, such as cancer and AIDS, longitudinal biomarkers are used to monitor disease progression and to predict survival. These longitudinal measures are often missing at failure times and may be prone to measurement errors. More importantly, time-dependent survival models that include the raw longitudinal measurements may lead to biased results. In previous studies these two types of data are frequently analyzed separately where a mixed effects model is used for the longitudinal data and a survival model is applied to the event outcome. Methods In this paper we compare joint maximum likelihood methods, a two-step approach and a time dependent covariate method that link longitudinal data to survival data with emphasis on using longitudinal measures to predict survival. We apply a Bayesian semi-parametric joint method and maximum likelihood joint method that maximizes the joint likelihood of the time-to-event and longitudinal measures. We also implement the Two-Step approach, which estimates random effects separately, and a classic Time Dependent Covariate Model. We use simulation studies to assess bias, accuracy, and coverage probabilities for the estimates of the link parameter that connects the longitudinal measures to survival times. Results Simulation results demonstrate that the Two-Step approach performed best at estimating the link parameter when variability in the longitudinal measure is low but is somewhat biased downwards when the variability is high. Bayesian semi-parametric and maximum likelihood joint methods yield higher link parameter estimates with low and high variability in the longitudinal measure. The Time Dependent Covariate method resulted in consistent underestimation of the link parameter. We illustrate these methods using data from the Framingham Heart Study in which lipid measurements and Myocardial Infarction data were collected over a period of 26 years. Conclusions Traditional methods for modeling longitudinal and survival data, such as the time dependent covariate method, that use the observed longitudinal data, tend to provide downwardly biased estimates. The two-step approach and joint models provide better estimates, although a comparison of these methods may depend on the underlying residual variance.

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Introduction

Cited by 46 publications

References 0 publications

Computing tails of compound distributions using direct numerical integration

Computing tails of compound distributions using direct numerical integration

Soluble Cytostatic Factor(s) Released from Human Monocytes

Revisiting methods for modeling longitudinal and survival data: Framingham Heart Study

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